MHB Proving $\sin\left(10^{\circ}\right)$ is Rational or Irrational

Click For Summary
The discussion centers on proving whether $\sin(10^{\circ})$ is rational or irrational. It utilizes the relation $\sin \alpha = 3\ \sin \frac{\alpha}{3} - 4\ \sin^{3} \frac{\alpha}{3}$, leading to the cubic equation $8\ x^{3} - 6\ x + 1 =0$. The argument posits that if $x$ were rational, it could be expressed as a fraction of integers, but this leads to a contradiction involving the square root of a non-integer. Consequently, the conclusion drawn is that $\sin(10^{\circ})$ is irrational. The discussion is supported by mathematical reasoning and a shared solution approach.
Fallen Angel
Messages
202
Reaction score
0
Is $\sin\left(10^{\circ}\right)$ rational or not? Prove it.
 
Mathematics news on Phys.org
Fallen Angel said:
Is $\sin\left(10^{\circ}\right)$ rational or not? Prove it.

[sp]From the relation...

$\displaystyle \sin \alpha = 3\ \sin \frac{\alpha}{3} - 4\ \sin^{3} \frac{\alpha}{3}\ (1)$

... it follows that $\displaystyle x= \sin \frac{\pi}{18}$ must be root of the cubic equation...

$\displaystyle 8\ x^{3} - 6\ x + 1 =0\ (2)$

If x is rational, then You can write $\displaystyle x = \frac{a}{b}$, being a and b integers... but in this case for (2) it must be...

$\displaystyle b = 4\ \sqrt{3\ a - \frac{1}{2}}\ (3)$

... and that's impossible for a and b integers... the conclusion is that x isn't rational...[/sp]

Kind regards

$\chi$ $\sigma$
 
Good work chisigma, my solution was almost the same.

From $8x^3-6x+1=0$, let $y=2 sin \ 10º$, then $y^3-3y+1=0$ and this polynomial has no rational roots (because of Gauss Lemma).
 

Thread 'Area of Overlapping Squares'

Here is a little puzzle from the book 100 Geometric Games by Pierre Berloquin. The side of a small square is one meter long and the side of a larger square one and a half meters long. One vertex of the large square is at the center of the small square. The side of the large square cuts two sides of the small square into one- third parts and two-thirds parts. What is the area where the squares overlap?
View full post »

Similar threads

Undergrad Rational powers of irrational numbers
  • · Replies 27 ·
Replies
27
Views
3K
MHB Show that sin10∘ is irrational number
  • · Replies 8 ·
Replies
8
Views
2K
MHB How Can Rational and Irrational Numbers Combine in Arithmetic Operations?
  • · Replies 16 ·
Replies
16
Views
3K
MHB Is (27/4)/(6.75) a Whole, Natural, Integer, Rational, or Irrational Number?
  • · Replies 4 ·
Replies
4
Views
3K
Undergrad Rational sequence converging to irrational
  • · Replies 13 ·
Replies
13
Views
9K
MHB We can find two irrational numbers x and y to make xy rational,true or false
  • · Replies 4 ·
Replies
4
Views
3K
MHB Irrational Raised to Irrational
  • · Replies 2 ·
Replies
2
Views
2K
MHB Rational Number: Proving $x+\dfrac{1}{x}$ is Rational
  • · Replies 1 ·
Replies
1
Views
1K
High School Proving the area formula for a rectangle for all positive real numbers
  • · Replies 10 ·
Replies
10
Views
2K
Irrational^rational = rational
  • · Replies 10 ·
Replies
10
Views
2K